Measurement Model Scaling and Identification

Conventional Drawing and Labeling of SEM Model

SEM nomenclature conventions

name	upper case	lower case	usage
Lambda	\(\Lambda\)	\(\lambda\)	Loading of a manifest indicator onto a latent construct
Psi	\(\Psi\)	\(\psi\)	residual variance/covariance of contruct when endogenous
Theta	\(\Theta\)	\(\theta\)	residual variance/covariance of indicators
Sigma	\(\Sigma\)	\(\sigma\)	\(\Sigma\) is the model implied variance/covariance matrix;
			\(\sigma\) is standard deviation, \(\sigma^2\) variance of indicator.
			\(\sigma\) can also be covariance of indicator

\[ \textbf{$\Sigma$} = \left[ \begin{array}{cccccc} \sigma_{1,1}^2 & \sigma_{1,2} & \sigma_{1,3} \\ \sigma_{2,1} & \sigma_{2,2}^2 & \sigma_{2,3} \\ \sigma_{3,1} & \sigma_{3,2} & \sigma_{3,3}^2 \end{array} \right],\]

\[ \textbf{$\Lambda$} = \left[ \begin{array}{cc} \lambda_{1,1} \\ \lambda_{2,1} \\ \lambda_{3,1} \end{array} \right],\]

\[ \textbf{$\Psi$} = \left[ \begin{array}{cc} \psi_{1,1} \end{array} \right],\]

\[ \textbf{$\Lambda^\prime$} = \left[ \begin{array}{cc} \lambda_{1,1} & \lambda_{2,1} & \lambda_{3,1} \end{array} \right],\]

\[ \textbf{$\Theta$} = \left[ \begin{array}{cccccc} \theta_{1,1} & 0 & 0 \\ 0 & \theta_{2,2} & 0 \\ 0 & 0 & \theta_{3,3} \end{array} \right].\]

Fundamental SEM equation

\[ \Sigma = \Lambda \Psi \Lambda' + \Theta \tag{1} \]

Measurement Model: 3 indicators

library(lavaan)

##Prepare data with sufficient statisitics##
mymeans<-matrix(c(3.06893, 2.92590, 3.11013), ncol=3,nrow=1)
mysd<-c(0.84194,0.88934,0.83470)
mat <- c(1.00000,
         0.55226, 1.00000,
         0.56256, 0.60307, 1.00000)
mycor <- getCov(mat, lower = TRUE)

##Transform correlation matrix to covariance matrix using information above##
mycov <- mysd %*% t(mysd)

rownames(mycor) <-c( "Glad", "Cheerful", "Happy")
colnames(mycor) <-c( "Glad", "Cheerful", "Happy")

rownames(mycov) <-c( "Glad", "Cheerful", "Happy")
colnames(mycov) <-c( "Glad", "Cheerful", "Happy")
mynob<-823

Descriptive Statistics

	Glad	Cheerful	Happy
Glad	1.00	0.55	0.56
Cheerful	0.55	1.00	0.60
Happy	0.56	0.60	1.00
Mean	3.07	2.93	3.11
SD	0.84	0.89	0.83
Var	0.71	0.79	0.70

Mplus input file (marker1.inp)

 [1] TITLE: Positive Affect with Marker Variable Scaling    
 [2] DATA:                                                  
 [3]     FILE IS PAcorr2t.dat;                              
 [4]     TYPE IS CORRELATION;                               
 [5]     NOBSERVATIONS ARE 823;                             
 [6] VARIABLE:                                              
 [7]      NAMES ARE glad1 cheer1 happy1 glad2 cheer2 happy2;
 [8]      usevariables are glad1-happy1;                    
 [9] ANALYSIS:                                              
[10]                                                        
[11] MODEL:                                                 
[12]     Positive BY glad1 cheer1 happy1;                   
[13] OUTPUT: stdyx;                                         

Mplus output using Correlation Table and first indicator fixed to 1.00

 [1] MODEL RESULTS                                                 
 [2]                                                     Two-Tailed
 [3]                     Estimate       S.E.  Est./S.E.    P-Value 
 [4]  POSITIVE BY                                                  
 [5]     GLAD1              1.000      0.000    999.000    999.000 
 [6]     CHEER1             1.072      0.061     17.612      0.000 
 [7]     HAPPY1             1.092      0.062     17.624      0.000 
 [8]  Variances                                                    
 [9]     POSITIVE           0.515      0.049     10.521      0.000 
[10]  Residual Variances                                           
[11]     GLAD1              0.484      0.033     14.547      0.000 
[12]     CHEER1             0.408      0.033     12.211      0.000 
[13]     HAPPY1             0.385      0.034     11.484      0.000 

lambda = matrix(c(1.00, 1.072, 1.092), nrow = 3)
lambda

      [,1]
[1,] 1.000
[2,] 1.072
[3,] 1.092

psi    = matrix(.515)
psi

      [,1]
[1,] 0.515

t(lambda)

     [,1]  [,2]  [,3]
[1,]    1 1.072 1.092

theta  = diag(c(.484, .408, .385))
theta

      [,1]  [,2]  [,3]
[1,] 0.484 0.000 0.000
[2,] 0.000 0.408 0.000
[3,] 0.000 0.000 0.385

sigma = lambda %*% psi %*% t(lambda) + theta
round(sigma,2)

     [,1] [,2] [,3]
[1,] 1.00 0.55 0.56
[2,] 0.55 1.00 0.60
[3,] 0.56 0.60 1.00

	Glad	Cheerful	Happy
Glad	1.00	0.55	0.56
Cheerful	0.55	1.00	0.60
Happy	0.56	0.60	1.00

Measurement Model: 6 indicators, 2 latent constructs

\[ \textbf{$\Sigma$} = \left[ \begin{array}{cccccc} \sigma_{1,1}^2 & \sigma_{1,2} & \sigma_{1,3} & \sigma_{1,4} & \sigma_{1,5} & \sigma_{1,6} \\ \sigma_{2,1} & \sigma_{2,2}^2 & \sigma_{2,3} & \sigma_{2,4} & \sigma_{2,5} & \sigma_{2,6} \\ \sigma_{3,1} & \sigma_{3,2} & \sigma_{3,3}^2 & \sigma_{3,4} & \sigma_{3,5} & \sigma_{3,6} \\ \sigma_{4,1} & \sigma_{4,2} & \sigma_{4,3} & \sigma_{4,4}^2 & \sigma_{4,5} & \sigma_{4,6} \\ \sigma_{5,1} & \sigma_{5,2} & \sigma_{5,3} & \sigma_{5,4} & \sigma_{5,5}^2 & \sigma_{5,6} \\ \sigma_{6,1} & \sigma_{6,2} & \sigma_{6,3} & \sigma_{6,4} & \sigma_{6,5} & \sigma_{6,6}^2 \end{array} \right],\]

\[ \textbf{$\Lambda$} = \left[ \begin{array}{cc} \lambda_{1,1} & 0 \\ \lambda_{2,1} & 0 \\ \lambda_{3,1} & 0 \\ 0 & \lambda_{4,2} \\ 0 & \lambda_{5,2} \\ 0 & \lambda_{6,2} \end{array} \right],\]

\[ \textbf{$\Psi$} = \left[ \begin{array}{cc} \psi_{1,1} & \psi_{1,2} \\ \psi_{2,1} & \psi_{2,2} \end{array} \right],\]

\[ \textbf{$\Lambda^\prime$} = \left[ \begin{array}{cc} \lambda_{1,1} & \lambda_{2,1} & \lambda_{3,1} 0 & 0 & 0\\ 0 & 0 & 0 & \lambda_{4,2} & \lambda_{5,2} & \lambda_{6,2} \end{array} \right],\]

\[ \textbf{$\Theta$} = \left[ \begin{array}{cccccc} \theta_{1,1} & 0 & 0 & 0 & 0 & 0 \\ 0 & \theta_{2,2} & 0 & 0 & 0 & 0 \\ 0 & 0 & \theta_{3,3} & 0 & 0 & 0 \\ 0 & 0 & 0 & \theta_{4,4} & 0 & 0 \\ 0 & 0 & 0 & 0 & \theta_{5,5} & 0 \\ 0 & 0 & 0 & 0 & 0 & \theta_{6,6} \end{array} \right].\]

lambda = matrix(c(.712, .788, .768, 0.0, 0.0, 0.0,
                  0.0, 0.0, 0.0, .729, .764, .778), nrow = 6)
lambda

      [,1]  [,2]
[1,] 0.712 0.000
[2,] 0.788 0.000
[3,] 0.768 0.000
[4,] 0.000 0.729
[5,] 0.000 0.764
[6,] 0.000 0.778

psi    = matrix(c(1.00, 0.561,
                  0.561, 1.00), nrow = 2)
psi

      [,1]  [,2]
[1,] 1.000 0.561
[2,] 0.561 1.000

theta = diag(c(.491, .378, .409, .467, .416, .394))
theta

      [,1]  [,2]  [,3]  [,4]  [,5]  [,6]
[1,] 0.491 0.000 0.000 0.000 0.000 0.000
[2,] 0.000 0.378 0.000 0.000 0.000 0.000
[3,] 0.000 0.000 0.409 0.000 0.000 0.000
[4,] 0.000 0.000 0.000 0.467 0.000 0.000
[5,] 0.000 0.000 0.000 0.000 0.416 0.000
[6,] 0.000 0.000 0.000 0.000 0.000 0.394

sigma = lambda %*% psi %*% t(lambda) + theta
round(sigma, 2)

     [,1] [,2] [,3] [,4] [,5] [,6]
[1,] 1.00 0.56 0.55 0.29 0.31 0.31
[2,] 0.56 1.00 0.61 0.32 0.34 0.34
[3,] 0.55 0.61 1.00 0.31 0.33 0.34
[4,] 0.29 0.32 0.31 1.00 0.56 0.57
[5,] 0.31 0.34 0.33 0.56 1.00 0.59
[6,] 0.31 0.34 0.34 0.57 0.59 1.00

vec <- dput(scan('topics/2_MeasurementModel/2b_ConfirmatoryFactorAnalysis/mplus/PAcorr2t.dat', quiet = TRUE))

c(1, 0.55226, 1, 0.56256, 0.60307, 1, 0.31889, 0.35898, 0.27757, 1, 0.24363, 0.35798, 0.31889, 0.56014, 1, 0.32217, 0.36385, 0.32072, 0.56164, 0.59738, 1)

mat <- getCov(vec, lower = TRUE)
ztable(mat)

	V1	V2	V3	V4	V5	V6
V1	1.00	0.55	0.56	0.32	0.24	0.32
V2	0.55	1.00	0.60	0.36	0.36	0.36
V3	0.56	0.60	1.00	0.28	0.32	0.32
V4	0.32	0.36	0.28	1.00	0.56	0.56
V5	0.24	0.36	0.32	0.56	1.00	0.60
V6	0.32	0.36	0.32	0.56	0.60	1.00

Adding Means to the Model

The means structure equation:

\[ \mu_y = \textbf{T} + \Lambda \textbf{A}. \tag{2} \]

Mplus: factorMeans.inp

 [1] TITLE: Positive Affect with Marker Variable Scaling and Mean Structure
 [2] DATA:                                                                 
 [3]     FILE IS PAmeansdcorr2t.dat;                                       
 [4]     TYPE IS MEANS STDEVIATIONS CORRELATION;                           
 [5]     NOBSERVATIONS ARE 823;                                            
 [6] VARIABLE:                                                             
 [7]      NAMES ARE glad1 cheer1 happy1 glad2 cheer2 happy2;               
 [8]      usevariables are glad1-happy1;                                   
 [9] ANALYSIS:                                                             
[10]                                                                       
[11] MODEL:                                                                
[12]     PA BY glad1* cheer1 happy1;                                       
[13]     PA@1;                                                             
[14]     [PA@0];                                                           

Mplus: factorMeans.out

 [1]                                                     Two-Tailed
 [2]                     Estimate       S.E.  Est./S.E.    P-Value 
 [3]  PA       BY                                                  
 [4]     GLAD1              0.604      0.029     21.041      0.000 
 [5]     CHEER1             0.684      0.030     22.631      0.000 
 [6]     HAPPY1             0.654      0.028     23.074      0.000 
 [7]  Means                                                        
 [8]     PA                 0.000      0.000    999.000    999.000 
 [9]  Intercepts                                                   
[10]     GLAD1              3.069      0.029    104.631      0.000 
[11]     CHEER1             2.926      0.031     94.438      0.000 
[12]     HAPPY1             3.110      0.029    106.954      0.000 
[13]  Variances                                                    
[14]     PA                 1.000      0.000    999.000    999.000 
[15]  Residual Variances                                           
[16]     GLAD1              0.343      0.024     14.550      0.000 
[17]     CHEER1             0.322      0.026     12.209      0.000 
[18]     HAPPY1             0.268      0.023     11.483      0.000 

Longitudinal Measurement Model

Descriptive Statistics

	Glad1	Cheer1	Happy1	Glad2	Cheer2	Happy2
Glad1	0.71	0.41	0.40	0.23	0.19	0.23
Cheer1	0.41	0.79	0.45	0.27	0.29	0.27
Happy1	0.40	0.45	0.70	0.19	0.24	0.22
Glad2	0.23	0.27	0.19	0.71	0.43	0.40
Cheer2	0.19	0.29	0.24	0.43	0.83	0.46
Happy2	0.23	0.27	0.22	0.40	0.46	0.71
Mean	3.07	2.93	3.11	3.03	2.86	3.09
SD	0.84	0.89	0.83	0.84	0.91	0.84
Var	0.71	0.79	0.70	0.71	0.83	0.70

Mplus: effectCodingMeansLongitudinal.inp

 [1] TITLE:                                                       
 [2]     Effects coding with means and Longitudinal               
 [3]     with 2 timepoints                                        
 [4] DATA:                                                        
 [5]     FILE IS PAmeansdcorr2t.dat;                              
 [6]     TYPE IS MEANS STD CORRELATION;                           
 [7]     NOBSERVATIONS ARE 823;                                   
 [8] VARIABLE:                                                    
 [9]     NAMES ARE Glad1 Cheer1 Happy1 Glad2 Cheer2 Happy2;       
[10]     !Note: glad1 cheer1 and happy1 are from Time 1           
[11]     !glad2 cheer2 and happy2 are from time 2                 
[12]                                                              
[13]     USEVARIABLES ARE Glad1 Cheer1 Happy1 Glad2 Cheer2 Happy2;
[14] MODEL:                                                       
[15]     Pos1 BY Glad1*(L1)                                       
[16]            Cheer1(L2)                                        
[17]            Happy1(L3) ; !Label for constraints               
[18]     Pos2 BY Glad2*(L4)                                       
[19]            Cheer2(L5)                                        
[20]            Happy2(L6);                                       
[21]     [Pos1]; !estimate latent means                           
[22]     [Pos2];                                                  
[23]     Glad1 with Glad2; !allow correlated residuals across time
[24]     Cheer1 with Cheer2;                                      
[25]     Happy1 with Happy2;                                      
[26]     [Glad1](T1)                                              
[27]     [Cheer1](T2)                                             
[28]     [Happy1](T3); !estimate and constrain the intercepts     
[29]     [Glad2](T4)                                              
[30]     [Cheer2](T5)                                             
[31]     [Happy2](T6);                                            
[32]     MODEL CONSTRAINT: L1= 3 - L2 - L3;                       
[33]                       L4 = 3 - L5 - L6;                      
[34]                       T1= 0 - T2 - T3;                       
[35]                       T4 = 0 - T5 - T6;                      
[36] OUTPUT:                                                      
[37]     TECH1;                                                   
[38]     RESIDUAL;                                                
[39]     STANDARDIZED;                                            

Mplus: effectcodingmeanslongitudinal.out

 [1] MODEL RESULTS                                                 
 [2]                                                     Two-Tailed
 [3]                     Estimate       S.E.  Est./S.E.    P-Value 
 [4]  POS1     BY                                                  
 [5]     GLAD1              0.925      0.031     29.969      0.000 
 [6]     CHEER1             1.075      0.032     33.613      0.000 
 [7]     HAPPY1             0.999      0.031     32.008      0.000 
 [8]  POS2     BY                                                  
 [9]     GLAD2              0.934      0.031     30.394      0.000 
[10]     CHEER2             1.061      0.032     33.062      0.000 
[11]     HAPPY2             1.004      0.031     32.116      0.000 
[12]  POS2     WITH                                                
[13]     POS1               0.234      0.020     11.509      0.000 
[14]  GLAD1    WITH                                                
[15]     GLAD2              0.031      0.015      2.055      0.040 
[16]  CHEER1   WITH                                                
[17]     CHEER2             0.018      0.016      1.076      0.282 
[18]  HAPPY1   WITH                                                
[19]     HAPPY2            -0.011      0.014     -0.767      0.443 
[20]  Means                                                        
[21]     POS1               3.035      0.025    120.379      0.000 
[22]     POS2               2.992      0.025    117.661      0.000 
[23]  Intercepts                                                   
[24]     GLAD1              0.260      0.095      2.737      0.006 
[25]     CHEER1            -0.338      0.098     -3.435      0.001 
[26]     HAPPY1             0.078      0.096      0.808      0.419 
[27]     GLAD2              0.231      0.093      2.469      0.014 
[28]     CHEER2            -0.319      0.097     -3.275      0.001 
[29]     HAPPY2             0.088      0.095      0.933      0.351 
[30]  Variances                                                    
[31]     POS1               0.420      0.026     16.126      0.000 
[32]     POS2               0.426      0.026     16.106      0.000 
[33]  Residual Variances                                           
[34]     GLAD1              0.349      0.023     15.214      0.000 
[35]     CHEER1             0.306      0.025     12.199      0.000 
[36]     HAPPY1             0.278      0.022     12.455      0.000 
[37]     GLAD2              0.335      0.023     14.746      0.000 
[38]     CHEER2             0.343      0.026     13.084      0.000 
[39]     HAPPY2             0.274      0.023     12.153      0.000 

lambda <- matrix(c(.925, 1.075, .999, 0, 0, 0,
                   0, 0, 0, .934, 1.061, 1.004), nrow = 6)
lambda

      [,1]  [,2]
[1,] 0.925 0.000
[2,] 1.075 0.000
[3,] 0.999 0.000
[4,] 0.000 0.934
[5,] 0.000 1.061
[6,] 0.000 1.004

psi <- matrix(c(.420, .234,
                .234, .426), nrow=2)
psi

      [,1]  [,2]
[1,] 0.420 0.234
[2,] 0.234 0.426

t(lambda)

      [,1]  [,2]  [,3]  [,4]  [,5]  [,6]
[1,] 0.925 1.075 0.999 0.000 0.000 0.000
[2,] 0.000 0.000 0.000 0.934 1.061 1.004

theta <- diag(c(.349, .306, .278, .335, .343, .274))
theta[1 ,4] <- theta[4, 1] <- .031 
theta[2, 5] <- theta[5, 2] <- .018
theta[3, 6] <- theta[6, 3] <- -.011
theta

      [,1]  [,2]   [,3]  [,4]  [,5]   [,6]
[1,] 0.349 0.000  0.000 0.031 0.000  0.000
[2,] 0.000 0.306  0.000 0.000 0.018  0.000
[3,] 0.000 0.000  0.278 0.000 0.000 -0.011
[4,] 0.031 0.000  0.000 0.335 0.000  0.000
[5,] 0.000 0.018  0.000 0.000 0.343  0.000
[6,] 0.000 0.000 -0.011 0.000 0.000  0.274

tau <- matrix(c(.260, -.338, .078, .231, -.319, .088),nrow = 6)
tau

       [,1]
[1,]  0.260
[2,] -0.338
[3,]  0.078
[4,]  0.231
[5,] -0.319
[6,]  0.088

alpha <- matrix(c(3.035, 2.992), nrow=2)
alpha

      [,1]
[1,] 3.035
[2,] 2.992

sigma <- lambda %*% psi %*% t(lambda) + theta
sigma

          [,1]      [,2]      [,3]      [,4]      [,5]      [,6]
[1,] 0.7083625 0.4176375 0.3881115 0.2331643 0.2296535 0.2173158
[2,] 0.4176375 0.7913625 0.4510485 0.2349477 0.2848946 0.2525562
[3,] 0.3881115 0.4510485 0.6971604 0.2183374 0.2480257 0.2237011
[4,] 0.2331643 0.2349477 0.2183374 0.7066237 0.4221549 0.3994755
[5,] 0.2296535 0.2848946 0.2480257 0.4221549 0.8225571 0.4537939
[6,] 0.2173158 0.2525562 0.2237011 0.3994755 0.4537939 0.7034148

mu <- tau + lambda %*% alpha
mu

         [,1]
[1,] 3.067375
[2,] 2.924625
[3,] 3.109965
[4,] 3.025528
[5,] 2.855512
[6,] 3.091968

Descriptive Statistics

	Glad1	Cheer1	Happy1	Glad2	Cheer2	Happy2
Glad1	0.71	0.41	0.40	0.23	0.19	0.23
Cheer1	0.41	0.79	0.45	0.27	0.29	0.27
Happy1	0.40	0.45	0.70	0.19	0.24	0.22
Glad2	0.23	0.27	0.19	0.71	0.43	0.40
Cheer2	0.19	0.29	0.24	0.43	0.83	0.46
Happy2	0.23	0.27	0.22	0.40	0.46	0.71
Mean	3.07	2.93	3.11	3.03	2.86	3.09
SD	0.84	0.89	0.83	0.84	0.91	0.84
Var	0.71	0.79	0.70	0.71	0.83	0.70